Applications of interval arithmetic for reduction of number of computations in ray tracing problems

ABSTRACT

Embodiments provide for ray tracing traversal that relies on selected geometrical properties of the application to reduce the number of operations required during each traversal step. The traversal algorithm does not depend on the number of rays in the group. As a result, multi-level traversal schemes may be implemented, starting with a large number of rays in a group and then reducing it as needed to maintain group coherency. Multi-level traversal schemes may be created by splitting large groups of rays while traversing acceleration structures.

BACKGROUND

Implementations of the claimed invention generally may relate to raytracing and, more particularly, to interval arithmetic for ray tracing.

Ray tracing is a well know method used in modeling of a variety ofphysical phenomena related to wave propagation in various media. Forexample it is used for computing an illumination solution inphotorealistic computer graphics, for complex environment channelmodeling in wireless communication, aureal rendering in advanced audioapplications, etc.

A ray is a half line of infinite length originating at a point in spacedescribed by a position vector which travels from said point along adirection vector. Ray tracing is used in computer graphics to determinevisibility by directing one or more rays from a vantage point describedby the ray's position vector along a line of sight described by theray's direction vector. To determine the nearest visible surface alongthat line of sight requires that the ray be effectively tested forintersection against all the geometry within the virtual scene andretain the nearest intersection.

When working with real values, data is often approximated byfloating-point (FP) numbers with limited precision. FP representationsare not uniform through the number space, and usually a desired realvalue (i.e. ⅓) is approximated by a value that is less than or greaterthan the desired value. The error introduced is often asymmetrical—thedifference between the exact value and the closest lower FPapproximation may be much greater or less than the difference to theclosest higher FP approximation. Such numerical errors may be propagatedand accumulate though all the computations, sometimes creating seriousproblems.

One way to handle such numerical inaccuracies is to use intervalsinstead of FP approximations. In this case, any real number isrepresented by 2 FP values: one is less than the real one, and anotheris greater than the real one. The bound values are preserved throughoutall computations, yielding an interval, which covers the exact solution.Usually, applications using interval arithmetic are limited to certainclasses of workloads (such as quality control, economics or quantummechanics) where the additional costs of such interval computationssignificantly outweigh the implications of dealing with inexact FPnumbers for any final values.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate one or more implementationsconsistent with the principles of the invention and, together with thedescription, explain such implementations. The drawings are notnecessarily to scale, the emphasis instead being placed uponillustrating the principles of the invention. In the drawings

FIG. 1 illustrates an example of multiple rays traced through a cellfrom a common origin, executed during traversal of a binary tree.

FIG. 2 illustrates an example of interval implementation of traversingmultiple rays through a binary tree.

FIG. 3 is a flow chart illustrating a process of traversing multiplerays through a binary tree using an interval technique.

DETAILED DESCRIPTION

The following detailed description refers to the accompanying drawings.The same reference numbers may be used in different drawings to identifythe same or similar elements. In the following description, for purposesof explanation and not limitation, specific details are set forth suchas particular structures, architectures, interfaces, techniques, etc. inorder to provide a thorough understanding of the various aspects of theclaimed invention. However, it will be apparent to those skilled in theart having the benefit of the present disclosure that the variousaspects of the invention claimed may be practiced in other examples thatdepart from these specific details. In certain instances, descriptionsof well known devices, circuits, and methods are omitted so as not toobscure the description of the present invention with unnecessarydetail.

Embodiments provide for ray tracing traversal that relies on selectedgeometrical properties of the application to reduce the number offloating point (or other data type operations such as integer, fixedpoint) operations required during each traversal step. The intervaltraversal algorithm does not depend on the number of rays in the group.Multi-level traversal schemes may be implemented, starting with a largenumber of rays in a group and then reducing it as needed to maintaingroup coherency. Additional rays may be generated during traversal toimprove anti-aliasing properties of the resulting image in areas of highgeometrical complexity. The interval traversal algorithm groups parallelgeometrical queries, extracts selected common geometricalcharacteristics pertinent for the whole group, and then executes a queryusing only these characteristics (and not the whole group). Ray tracingis based on massively parallel geometrical queries, executed againstsome spatially ordered geometrical database. The interval traversalalgorithm may be extended to cover other types of applications, where itmay be possible to find and trace certain group properties against aspecialized database. One skilled in the art will recognize thatembodiments of the invention are not limited to floating pointimplementation. Rather, the embodiments of the invention may beimplemented using various data types, including but not limited tointeger, fixed point and so forth.

FIG. 1 illustrates an example 100 of multiple rays 102 traced through acell 104 from a common origin 106 for traversal of a binary tree. Onecell 104 is split into two subspaces including a nearest cell C0 108 andfurthest cell C1 110 by split plane P0. Rays 102 may be shot throughpixels on the screen into a database representing all of the objects ina scene. These objects (suitably sub-divided) and data may representempty space, and may be stored in a hierarchical spatial partitioningstructure. Shooting a ray 102 involves tracing the path the ray 102 maytake through this structure. Opportunities for parallelism exist but arelimited, as each ray may take a different path through the database, andsince the data structure is hierarchical, there is a sequentialdependency as the ray goes from one level to the next.

The database may represent the distribution of objects and empty spaceas a collection of axis aligned spatial intervals. A collection of raysmay be directly tested against any arbitrary level of the databasehierarchy (i.e. not necessary starting at the top). The bundles of raysmay be subdivided proceeding down the structure.

This results in improved numerical fidelity and simplifies the processof tracing rays. In particular, the number of operations required perray is reduced, resulting in an improvement in overall applicationperformance. Furthermore, hardware may be designed to directly implementsuch interval arithmetic, allowing additional performance improvement.Shooting of rays is not particular to graphics, similar technology mayalso used to track the propagation of waves of various kinds,calculating radar cross sections for military purposes etc.

In a ray tracing environment, it may be required to shoot lots of rays.One way to accomplish this is to determine the intersection of all raysagainst all of the polygons that define all of the geometrical objectsin the scene.

Another way to accomplish this is to partition all of these polygonsinto an axis aligned partitioning structure. One implementation of thisis to split the entire scene up into a uniform grid of cubes, whilereplicating polygons that straddle the cube boundaries. A ray may beshot and the cubes the ray passes through predicted. The ray is onlytested against the contents of each of these cubes, ignoring the rest.Due to the relative efficiency of using such a representation versustesting every ray against every polygon, the term “accelerationstructure” may be used to describe any such data structure designed toreduce the total number of ray-polygon intersection tests.

The above uniform grid of cubes has the advantage that the trajectory ofa ray through the cubes may be calculated easily, and the relevant dataaccessed directly. The detail in the scene may not be distributed evenlythough. For example, a huge amount of polygons may end up in one cube,and very little detail in the others.

Another acceleration structure construct is commonly referred to as akd-tree. In this acceleration structure, some cost function may be usedto recursively split the scene by axis-aligned planes. Initially, thescene may be split in two by such a plane, each half may then be splitagain along some other plane, and so forth. This results in ahierarchical organization of the structure. Each level of theacceleration structure may be recursively traversed to determine wherethe next level of the structure can be found. Cost functions arecarefully chosen in the construction phase of these structures toachieve optimum performance while traversing these trees later whenshooting the various rays needed for visualization.

The leaf nodes of a kd-tree represent a small axis aligned cell whereinthere is some number of polygons. At the next level up the tree, eachnode represents an axis aligned box which is completely filled by two ofthe leaf nodes (a “split-plane” splits the larger volume into the twoleaf cells). At the next level, each node represents an axis aligned boxcompletely filled by two of the lower level nodes using a similarsplit-plane and so on. The tree is not required to be balanced, that isany internal node may be split into leaf node and another internal node.At any given level, a ray may be intersected against the bounding box todetermine whether: (1) the ray completely misses the box, (2) the rayhits the box and passes through the “left” sub-node—i.e. to the “left”of the split-plane, (3) the ray hits the box and passes through the“right” sub-box, or (4) the ray hits and passes through both of thesub-boxes. In the first case (1), the further processing of the lowerlevel nodes is no longer necessary, as the ray “misses” the entire lowerpart of the tree.

Embodiments of the invention are applicable to many accelerationstructures, including those that use separation planes to determinewhich objects have to be tested for a particular ray. These accelerationstructures include but are not limited to grids, bounding boxes andkd-trees.

Referring to FIG. 1, For any given block of rays, the traversalalgorithms determine if the rays 102:

-   (1) pass through subspace 108 to the left of the split-plane 104;-   (2) pass through the subspace 110 to the right of the split-plane    104; or-   (3) passes through both sub-spaces 108 and 110.

During full traversal of a binary tree, for each ray 102, the cell entryand exit points are known. These are the distances represented by oa,ob, oc, od, and oA, oB, oC, oD which are known from previouscomputations. The intersection points with the split-plane P₀ arecalculated. They are represented as distances oα, oβ, oχ, and oδ. Entryand exit distances are compared with the plane intersection. Forexample, referring to FIG. 1, rays oa and ob will go only through theleft cell 108, while rays oc and od go through both cells 108 and 110.The process is repeated for each subsequent cell that the rays passthrough.

If the algorithm requires ray traversal of both cells 108 and 110, thenall information, pertinent to the furthest cell such as 110, is storedin a stack-like structure. It includes, in particular, the distances tothe entry points oχ and oδ and the exit points oC and oD. The nearestcell 108 is recursively traversed first by executing all of the steps ofthe current processs with entry points a, b, c, and d and exit points A,B, χ, and δ. Once all cells within the nearest one have been traversed,the furthest cell data 110 is retrieved from the stack and the wholeprocess is repeated.

If some cell contains primitive objects (such as triangles), theremaining rays which pass through this cell are tested against theseobjects. For example, ray/triangle intersection tests are performed.

In some cases, for each ray, a primitive object has been discovered suchthat the distance to it is less than the distance to the current cell.In this case, subsequent traversal steps are not necessary. If raytracing is being used for rendering purposes, this refinement may beused if such a primitive object is opaque.

FIG. 2 illustrates an example 200 of multiple rays 202 traced through acell 204 from a common origin 206 for interval traversal of a binarytree. FIG. 2 shows the next traversal step after the cell is split. Inparticular, FIG. 1 shows the case where one cell is split into twosubspaces C₀ 108 and C₁ 110 by split-plane P₀. In FIG. 2, C₀ 208 isfurther split into C₀₀ 210 and the union of C₂ 212 and C₃ 214. No ray202 intersects the nearer sub-space C₀₀ 210.

The interval traversal algorithm is built upon the calculation andmaintenance of one single interval for a group of rays, which includesminimum and maximum distances for all the rays in the bunch from aselected point (camera position) to a particular cell. Instead ofrepresenting individual rays as 3D vectors pointing in particulardirections, a collection of rays may be represented as a single 3Dvector of intervals pointing approximately in some particular direction.Typically, the more coherent these rays are, the tighter the intervalsmay be. For each coordinate x, y, and z, this interval may be defined asminimum and maximum coordinate value among all rays. Similarly, theindividual cells of the acceleration structure may be represented asintervals in x, y and z. Cells at any level of a hierarchicalacceleration structure may be represented as such an interval. Upontraversing deeper into the acceleration structure, the vector ofintervals representing one group of rays may be sub-divided intomultiple groups of rays for efficiency. Higher degrees of ray coherencyare typically found deeper in the acceleration structure.

FIG. 3 is a flow chart illustrating a process 300 of interval traversalof a binary tree. Although process 300 may be described with regard toFIG. 2 for ease of explanation, the claimed invention is not limited inthis regard. Acts 302, 304, and 306 of the FIG. 3 are executed once pertraversal step, while acts 308-316 are executed for each traversed cell.

In act 302, a group of rays is generated and some common characteristicsof the group of rays are computed. For those rays generated from aparticular common point of origin such as camera position o through ascreen with pixels p_(xy), the following are computed for eachcoordinate axis:

In act 304, the minimum and maximum distance values among allprojections of direction vectors op_(xy) on any given axis are computed.By definition, for every ray in the group, the x, y, and z coordinatesof op_(xy) vector will be inside an appropriate interval. At thebeginning of the top cell traversal (act 304), the minimum and maximumdistances oa₁ and oA₁ are determined. These may be designated asinterval [oa₁, oA₁]. This interval is maintained and potentiallymodified (narrowed) during the remaining traversal process. Bydefinition, for any ray in the group, the distance to the nearest cellentry point is not less than oa₁ and the distance to the furthest cellexit point is less or equal to oA₁.

In act 306, inverse direction intervals are defined.

In act 308, the minimum and maximum distances to the split planeod_(min) and od_(max) may be computed using inverse direction intervalsdefined in act 306.

As shown in FIG. 2, sub-cells C₂ 212 and C₃ 214 are split by plane P₂.It is determined whether both sub-cells C₂ 212 and C₃ 214 are traversed.In particular, this is determined by evaluating the following twoconditions, which if satisfied, result in traversing only one sub-cell:

In act 310, if the minimum distance to the cell (oa₁) is more thanmaximum distance to the plane (oA₂), the [oa1, oA1] interval is modifiedand only the right sub-cell is traversed (act 312).

In act 314, if the maximum distance to the cell (oA₁) is less thanminimum distance to the plane (oa₃), the [oa1, oA1] interval is modifiedand only the left sub-cell is traversed (act 316).

If neither of these conditions are true, both sub-cells have to betraversed (act 318) and appropriate intervals have to be modified. Asshown in FIG. 2, during C₂ traversal, the interval will be [oa₁, oA₂].For the cell C₃, it will be [oa₃, oA₁].

One skilled in the art will recognize that different implementations ofthe interval traversal embodiments described herein are possible. Forexample, embodiments described can be extended to ray groups which donot have a common origin. Although process 300 may be implemented onmodern vector or SIMD (Single Instruction Multiple Data) machines, theclaimed invention is not limited in this regard.

Certainly, different implementations of the interval traversal algorithmare possible. One, provided above, is used only for presentationpurposes, as well as particular cases featured on the supplied figures.It is also possible to extend the ideas, outlined here, to a moregeneral case of ray bunches, which do not have common origin. Thefollowing observation helps to understand the differences between thefull and the interval traversal algorithms. The full algorithm basicallyimplements simultaneous bounding box clipping of a particular group ofrays. For any given cell, reached in the acceleration structure, theentry and exit points for all rays are known. The interval algorithmshown in FIG. 3 represents a lazy distributed box clipping, yieldingguaranteed minimum and maximum clipping distances for the whole group ofrays.

Embodiments of the invention may sharply reduce the number of floatingpoint or other data type operations required during each traversal step.Unlike the full traversal algorithm, the interval traversal algorithmdoes not depend on the number of rays in the group. Multi-leveltraversal schemes may be implemented starting with a large number ofrays in a group and then reducing it as needed to maintain groupcoherency. The interval traversal algorithm, if implemented or supportedin hardware, may enable a sharp reduction of power, consumed by thedevice, as well as increasing overall performance. Ray tracing is basedon massively parallel geometrical queries, executed against somespatially ordered geometrical database. The interval traversal algorithmconsists of grouping such queries, extracting certain common geometricalcharacteristics, pertinent for the whole group, and then executing aquery using only these characteristics (and not the whole group). Assuch, the interval traversal approach may be extended to cover othertypes of applications, where it may be possible to find and tracecertain group properties against a specialized database.

Although systems are illustrated as including discrete components, thesecomponents may be implemented in hardware, software/firmware, or somecombination thereof. When implemented in hardware, some components ofsystems may be combined in a certain chip or device.

Although several exemplary implementations have been discussed, theclaimed invention should not be limited to those explicitly mentioned,but instead should encompass any device or interface including more thanone processor capable of processing, transmitting, outputting, orstoring information. Processes may be implemented, for example, insoftware that may be executed by processors or another portion of localsystem.

The foregoing description of one or more implementations consistent withthe principles of the invention provides illustration and description,but is not intended to be exhaustive or to limit the scope of theinvention to the precise form disclosed. Modifications and variationsare possible in light of the above teachings or may be acquired frompractice of various implementations of the invention.

No element, act, or instruction used in the description of the presentapplication should be construed as critical or essential to theinvention unless explicitly described as such. Also, as used herein, thearticle “a” is intended to include one or more items. Variations andmodifications may be made to the above-described implementation(s) ofthe claimed invention without departing substantially from the spiritand principles of the invention. All such modifications and variationsare intended to be included herein within the scope of this disclosureand protected by the following claims.

1. A computer-implemented method for ray tracing, comprising: generatinga group of rays having a common point of origin; computing minimum andmaximum distances values among all projections of direction vectors onany given axis; defining inverse direction intervals representative ofthe group of rays; computing minimum and maximum distances to a splitplane using inverse direction intervals; determining whether sub-cellsin a split plane are traversed, comprising: modifying the interval andtraversing only one sub cell if the minimum distance to the cell is morethan the maximum distance to the plane or the maximum distance to thecell is less then the minimum distance to the plane.
 2. Thecomputer-implemented method claimed in claim 1, wherein modifying theinterval and traversing only one sub cell if the minimum distance to thecell is more than the maximum distance to the plane or the maximumdistance to the cell is less then the minimum distance to the planefurther comprises: in response to the minimum distance to the cell beingmore than maximum distance to the plane, modifying the interval andtraversing only a first sub-cell.
 3. The computer-implemented methodclaimed in claim 1, wherein modifying the interval and traversing onlyone sub cell if the minimum distance to the cell is more than themaximum distance to the plane or the maximum distance to the cell isless then the minimum distance to the plane further comprises: inresponse to the maximum distance to the cell being less than minimumdistance to the plane, modifying the interval and traversing only asecond sub-cell.
 4. The computer-implemented method claimed in claim 1,further comprising: modifying intervals and traversing both sub cells ifneither condition is met.
 5. A tangible machine-accessible mediumincluding instructions that, when executed, cause a machine to: generatea group of rays having a common point of origin; compute minimum andmaximum distances values among all projections of direction vectors onany given axis; define inverse direction intervals representative of thegroup of rays; compute minimum and maximum distances to a split planeusing inverse direction intervals; determine whether sub-cells in asplit plane are traversed, comprising: modify the interval and traversea cell if the minimum distance to the cell is more than the maximumdistance to the plane or the maximum distance to the cell is less thenthe minimum distance to the plane.
 6. An apparatus, comprising: adatabase; and a controller for generating a group of rays having acommon point of origin, computing minimum and maximum distances valuesamong all projections of direction vectors on any given axis, defininginverse direction intervals representative of the group of rays,computing minimum and maximum distances to a split plane using inversedirection intervals, determining whether sub-cells in a split plane aretraversed, comprising modifying the interval and traversing a cell ifthe minimum distance to the cell is more than the maxsmum distance tothe plane or the maximum distance to the cell is less then the minimumdistance to the plane.
 7. An system, comprising: a memory including adatabase; and a controller for generating a group of rays having acommon point of origin, computing minimum and maximum distances valuesamong all projections of direction vectors on any given axis, defininginverse direction intervals representative of the group of rays,computing minimum and maximum distances to a split plane using inversedirection intervals, determining whether sub-cells in a split plane aretraversed, comprising modifying the interval and traversing a cell ifthe minimum distance to the cell is more than the maximum distance tothe plane or the maximum distance to the cell is less then the minimumdistance to the plane.
 8. A computer-implemented method for ray tracing,comprising: generating a group of rays having a common point of origin:computing minimum and maximum distances values among all projections ofdirection vectors on any given axis; defining inverse directionintervals representative of the group of rays; computing minimum andmaximum distances to a split plane using inverse direction intervals;determining whether sub-cells in a split plane are traversed,comprising: modifying the interval and traversing a cell if the minimumdistance to the cell is more than the maximum distance to the plane orthe maximum distance to the cell is less then the minimum distance tothe plane.
 9. The computer-implemented method claimed in claim 8,wherein modifying the interval and traversing a cell if the minimumdistance to the cell is more than the maximum distance to the plane orthe maximum distance to the cell is less then the minimum distance tothe plane further comprises: in response to the minimum distance to thecell being more than maximum distance to the plane, modifying theinterval and traversing only a first sub-cell.
 10. Thecomputer-implemented method claimed in claim 8, wherein modifying theinterval and traversing a cell if the minimum distance to the cell ismore than the maximum distance to the plane or the maximum distance tothe cell is less then the minimum distance to the plane furthercomprises: in response to the maximum distance to the cell being lessthan minimum distance to the plane, modifying the interval andtraversing only a second sub-cell.
 11. The computer-implemented methodclaimed in claim 8, further comprising: modifying intervals andtraversing both sub cells if neither condition is met.